The complexity of the gravitational equations has led me to interests in other sorts of nonlinear physics and in general methods for studying them, the most common being infinite-dimensional Lie algebras of local and/or non-local symmetries of the associated nonlinear PDE's.

The main interest is in solutions to Einstein's field equations, and in perturbations of both the solutions themselves and massless field equations defined over the corresponding curved spaces. The approach to solutions is physical in motivation but mathematical in content. I often use the methods of complex differential manifolds, leading to the study of the structure of all possible self-dual spaces (H-spaces) with complex or Euclidean signature. A natural generalization of this is the study of HH-spaces, which may possess Minkowski signature, real, algebraically-degenerate cross-sections. Work on HH-spaces is proceeding by division into distinct Petrov types, leading particularly to the study of gravitational waves. Before his death much of this work was done in active collaboration with Jerzy Plebanski at the Instituto de Investigaciones y Estudios Avanzados del Instituto Politecnico Nacional in Mexico City and the University of Warsaw in Poland.

Some publications in this area are the following:

- "Third-order ODE's for Twisting, Type-N Vacuum Solutions,"
Class. Qu. Grav.
**11**(1994) 157-166, with J. F. Plebanski and M. Przanowski. - "The Involutive Prolongation of the (Complex) Twisting, Type-N Vacuum Field Equations," Proc. of the International Conf. of Aspects of General Relativity and Mathematical Physics, N. Breton, R. Capovilla and T. Matos (Eds.), published by CINVESTAV, Mexico City, 1994, with Andrew Price.
- "An iterative approach to twisting and diverging, type-N, vacuum
Einstein equations: a (third-order) resolution of Stephani's
`paradox'," Class. Qu. Grav.
**14**(1997) 489-497, with J. F. Plebanski and Maciej Przanowski. - "Killing-Vector Reductions for Complex-Valued, Twisting, Type-N
Vacuum Solutions," to be published in the proceedings of the
9th Marcel Grossmann Conference, held in Rome, July, 2000.
Published in
*Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity,*V.G. Gurzadyan, R.T. Jantzen, and Remo Ruffini (Eds.) (World Scientific, 2002), p. 839-841. Also available online in 2001.

The pdf-version is available here. - "Equations for Complex-Valued, Twisting, Type N, Vacuum Solutions, with one or two Killing/homothetic vectors," from a talk at GR16, Durban, South Africa, July, 2001. A pdf-version is here, which is also available from the archive.
- "Asymptotic properties of the C-metric," with Pavel Sladek (Charles University, Prague,
Czech Republic). Class. Qu. Grav.
**27**205020 (2010). A pdf-version is available here. - "Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry," with Xuefeng Zhang.
Class. Qu. Grav.
**29**(2012) 065010. A pdf-copy is available here. - "Interpretation of twisting type N vacuum solutions with cosmological constant," with Xuefeng Zhang.
Class. Qu. Grav.
**30**(2013) 075021. A pdf-copy is available here. - "CR structures and twisting vacuum spacetimes with two Killing vectors and cosmological
constant: type II and more special," with Xuefeng Zhang.
Class. Qu. Grav.
**30**(2013) 115006. A pdf-copy is available here.

Areas of investigation in nonlinear physics include methods of finding families of particular, exact solutions of many different systems of nonlinear equations, motivation for the choices being found in problems in hydrodynamic flow, chemical mixing, nonlinear optics, three-level laser systems, and gravitational waves. These methods are used to generate families of soliton or breather solutions or Bäcklund transformations between equations, and to establish connections with many (other) interesting mathematical structures, including complex manifold theory, twistor methods, infinite solvable algebras, Kac-Moody algebras, and larger algebras (of infinite growth). John K. McIver is a collaborator in this work.

Some publications in this area are the following:

- "Infinite-dimensional Estabrook-Wahlquist Prolongations for the
Sine-Gordon Equation," J. Math. Phys.
**36**(1995), 5707-5734, with John K. McIver. - "The Robinson-Trautman Type III Prolongation Structure Contains
*K*_{2}," Commun. Math. Phys.**178**(1996) 375-390. This identifies carefully an important, still-unsolved problem in both general relativity and in infinite-dimensional Lie algebras (of infinite growth). Although published, here is a pdf-version if you want. - "Estabrook-Wahlquist Prolongations and Infinite-Dimensional Algebras,"
*Symmetry Methods in Physics,*VII International Conference, Dubna, Russia, 1995, edited by A.N. Sissakian and G.S. Pogosyan, published by the Joint Institute for Nuclear Research, Dubna, 1996, p. 203-211.

A pdf-version may be found by clicking here. - "Twisting gravitational waves and eigenvector fields for SL(2,C)
on an infinite jet," Electron. J. Diff. Eqns.,
**Conf. 04,**(2000) 75-85. This brings together intimately the difficulties in the two different areas of interest.

This may be accessed on line, since this is an electronic journal, at the following URL http://www.ma.hw.ac.uk/EJDE/conf-proc/04/f2/abstr.html , or via an Acrobat-readable (*.pdf) file right here. - A sequence of papers concerning "progress" on the general solution
of the field equations for self-dual, vacuum Einstein spaces with one
(rotational) Killing vector,
usually called the SDiff(2) Toda equation:
- "Difficulties with the SDiff(2) Toda Equation," CRM Proceedings and
Lecture Notes (Proceedings of a Conference in Halifax, June, 1999),
vol. 29 (American Mathematical Society, 2001), p. 217-224.

The Acrobat-readable file is available here. - "Infinite-Dimensional Symmetry Algebras as a Help Toward Solutions
of the Self-Dual Field Equations with one Killing Vector," with John K.
McIver, in the proceedings of the 9th Marcel Grossmann
Conference, held in Rome, July, 2000. Published in
*Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity,*V.G. Gurzadyan, R.T. Jantzen, and Remo Ruffini (Eds.) (World Scientific, 2002), p. 871-879. Also available online in 2001.

The Acrobat-readable file is available here. - "Generalized Symmetries for the sDiff(2) Toda Equation," with John K. McIver, in the Proceedings
of the 2002 International Conference,
*Topics in Mathematical Physics, General Relativity and Cosmology, in Honor of Jerzy Plebañski*, H. Garca, B. Mielnik, M. Montesinos & M. Przanowski, Eds., World Scientific Publ. Co., Hackensack, N.J., 2006, p. 177-191. A pdf-version of the content of the talk can be found at this link, while this link provides the published format. - "Non-Abelian Infinite Algebra of Generalized
Symmetries for the SDiff(2)Toda Equation" does (finally)
provide some real progress with this
equation, describing the complete, non-Abelian algebra of generalized symmetries (or Lie-
Bäcklund transformations) for this equation. The link in the previous sentence is
our original version; this link provides a pdf-file
of the version published in J. Phys. A
**37**(2004) 5825-5847. - "Solutions of the sDiff(2)Toda equation with SU(2) Symmetry" provides the general solution of this equation when the algebra of Killing vectors is increased to be the Lie algebra for SU(2), which is the same as the algebra for SO(3). It was published as Class. Quantum Grav. 27 (2010) 145001, with abstract and connection to the complete paper at this link.

- "Difficulties with the SDiff(2) Toda Equation," CRM Proceedings and
Lecture Notes (Proceedings of a Conference in Halifax, June, 1999),
vol. 29 (American Mathematical Society, 2001), p. 217-224.

Back to Daniel Finley's Dept. Page | To Daniel Finley's Main Links Page |

Last updated/modified: 30 May, 2013